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Learn moreIntroduction: What is the Rule of Three?
The Rule of Three (known historically in India as Trairāsika) is a fundamental mathematical method used to solve problems involving proportional reasoning.
When four quantities are linked in a proportion, and three of these quantities are known, the Rule of Three allows us to compute the fourth, unknown quantity.
When do we use the Rule of Three?
Use the Rule of Three when:
\(✔\) Two quantities increase or decrease together in the same proportion.
\(✔\) Three quantities are known, and the fourth quantity is to be found.
\(✔\) The relationship between the quantities remains constant.
Foundation of Rule of Three:
Consider two proportional ratios \(a : b\) and \(c : d\).
Method A: The Scaling Factor Approach
If two ratios are proportional, the target quantities are obtained by multiplying the base quantities by a common factor of change \((f)\).
\(c = f \cdot a\) \(\implies f = \frac{c}{a}\)
\(d = f \cdot b\) \(\implies f = \frac{d}{b}\)
Equating the two expressions of \(f\):
\(\frac{c}{a} = \frac{d}{b}\)
Method B: Cross Multiplication
\(\frac{c}{a} = \frac{d}{b}\)
Multiply both sides by \(a \cdot b\).
\(ab \times \frac{c}{a} = ab \times \frac{d}{b}\)
\(bc = ad\) or \(ad = bc\)
This is called Cross Multiplication.
If one value is unknown, then:
\(d = \frac{bc}{a}\)
This formula helps us find the missing quantity quickly.
The Product Rule of Proportion:
Two ratios are proportional if and only if the product of the extremes \((a \cdot d)\) equals the product of the means \((b \cdot c)\).
Trairāsika – An Ancient Indian Method
This method has been used in India since ancient times. Great mathematicians like Āryabhaṭa used it to solve practical problems involving trade, measurement, and daily life. Prominent mathematician Āryabhaṭa formulated this rule using four foundational terms:
| Sanskrit Term | Literal Meaning | Mathematical Role | Symbol |
| Pramāṇa | Measure / Base input | Known antecedent | \(a\) |
| Phala | Fruit / Result of the measure | Known consequent | \(b\) |
| Ichchhā | Requisition / Desired input | Second known antecedent | \(c\) |
| Ichchhāphala | Yield / Desired result | Unknown target quantity | \(d\) |
Āryabhaṭa’s Rule:
In Sanskrit mathematical tradition, the relationship is stated as:
Pramāṇa \(:\) Phala \(::\) Ichchhā \(:\) Ichchhāphala
Pramāṇa \(\times\) Ichchhāphala\(=\) Ichchhā \(\times\) Phala
\(\text{Ichchhāphala} = \frac{\text{Phala} \times \text{Ichchhā}}{\text{Pramāṇa}}\)
\(\implies d = \frac{b \times c}{a}\)
Using the cross-multiplication method proposed by Āryabhaṭa, ancient Indians solved complex problems involving proportionality.
Example:
A car travels \(90 \ km\) in \(150\) minutes. How far will it travel in \(4\) hours?
Solution:
First, convert to the same units.
\(4\) hours \(= 4 \times 60\) minutes
Now, write it in proportion.
\(150 : 90 :: 240 : x\)
Apply the rule of three,
\(d = \frac{bc}{a}\)
\(x = \frac{90 \times 240}{150}\)
\(x = 144\)
Distance travelled \(= 144 \ km\)
Before forming a proportion, both quantities must have the same units.
Important!
When Should We NOT Use the Rule of Three?
The Rule of Three works only for direct proportion.
Sometimes one quantity increases while the other decreases.
Sometimes one quantity increases while the other decreases.
Example: When speed increases, the time decreases
These quantities are not directly proportional, so the Rule of Three cannot be applied directly.
These quantities are not directly proportional, so the Rule of Three cannot be applied directly.
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